Browsing by Subject "Green function"

Now showing 1 - 2 of 2

Access status: Open Access ,
Existence of positive continuous weak solutions for some semilinear elliptic eigenvalue problems
(Wydawnictwa AGH, 2022) Zeddini, Noureddine; Sari, Rehab Saeed
Let $D$ be a bounded $C^{1,1}$-domain in $\mathbb{R}^d$, $d \geq 2$. The aim of this article is twofold. The first goal is to give a new characterization of the Kato class of functions $K(D)$ that was defined by N. Zeddini for $d=2$ and by H. Mâagli and M. Zribi for $d \geq 3$ and adapted to study some nonlinear elliptic problems in $D$. The second goal is to prove the existence of positive continuous weak solutions, having the global behavior of the associated homogeneous problem, for sufficiently small values of the nonnegative constants $\lambda$ and $\mu$ to the following system $\Delta u=\lambda f(x,u,v)$, $\Delta v=\mu g(x,u,v)$ in $D$, $u=\phi_{1}$ and $v=\phi_{2}$ on $\partial D$, where $\phi_{1}$ and $\phi_{2}$ are nontrivial nonnegative continuous functions on $\partial D$. The functions $f$ and $g$ are nonnegative and belong to a class of functions containing in particular all functions of the type $f(x,u,v)=p(x)u^{\alpha}h_{1}(v)$ and $g(x,u,v)=q(x)h_{2}(u)v^{\beta} with $\alpha \geq 1$, $\beta \geq 1$, $h_1$, $h_2$ are continuous on $[0,\infty)$ and $p$, $q$ are nonnegative functions in $K(D)$.
Access status: Open Access ,
Influence of an Lp-perturbation on Hardy-Sobolev inequality with singularity a curve
(Wydawnictwa AGH, 2021) Ijaodoro, Idowu Esther; Thiam, El Hadji Abdoulaye
We consider a bounded domain $\Omega$ of $\mathbb{R}^{N}$, $N \geq 3$, $h$ and $b$ continuous functions on $\Omega.$ Let $\Gamma$ be a closed curve contained in $\Omega$. We study existence of positive solutions $u \in H^{1}_{0}(\Omega)$ to the perturbed Hardy-Sobolev equation: $-\Delta u+hu+bu^{1+\delta}=\rho^{-\sigma}_{\Gamma} u^{2^*_{\sigma}-1} \quad \textrm{ in } \Omega,$ where $2^*_{\sigma}:=\frac{2(N-\sigma)}{N-2}$ is the critical Hardy-Sobolev exponent, $\sigma \in [0,2)$, $0\lt\delta\lt\frac{4}{N-2}$ and $\rho_{\Gamma}$ is the distance function to $\Gamma$. We show that the existence of minimizers does not depend on the local geometry of $\Gamma$ nor on the potential $h$. For $N=3$, the existence of ground-state solution may depends on the trace of the regular part of the Green function of $-\Delta+h$ and or on $b$. This is due to the perturbative term of order $1+\delta$.